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1) They drop the $$\cos \frac{(H\pi)}{\pi} \cdot \Delta^{H + 1/2}$$ 2) They divide by a normalization factor, which is the sum of the integrand (without the $\log v_s$). If you integrate: $$\cos \frac{(H \pi)}{\pi} \cdot \Delta^{H + 1/2} \cdot \frac{1}{(x + \Delta) \cdot x^{H + 1/2}}$$ from zero to infinity, you will get 1.
I am not going to answer all of your questions, but let me give it a go. I don't have a qualified answer to this one. In practice however $\beta$ is always bounded such that $\beta \in [0,1]$ (0 and 1 both included). see SABR chapter in Derman & Miller (2006). But as far I know, $\beta$ is not bounded in the original paper so in theory it can take any ...